LIFE OF MATHEMATICIAN G.CANTORLIFE OF MATHEMATICIAN G.CANTOR

LIFE OF MATHEMATICIAN R.DEDEKINDLIFE OF MATHEMATICIAN R.DEDEKIND

LIFE OF MATHEMATICIAN ARCHIMEDESLIFE OF MATHEMATICIAN ARCHIMEDES

Georg Ferdinand Ludwig Philipp Cantor ( March 3, 1845, St. Petersburg, Russia – Georg Ferdinand Ludwig Philipp Cantor ( March 3, 1845, St. Petersburg, Russia – January 6, 1918, Halle, Germany) was a German mathematician who is best known January 6, 1918, Halle, Germany) was a German mathematician who is best known as the creator of set theory. as the creator of set theory. He was born between 1809 and 1814 in He was born between 1809 and 1814 in CopenhagenCopenhagen, , DenmarkDenmark, and brought up in a , and brought up in a Lutheran Lutheran GermanGerman mission in St. Petersburg. mission in St. Petersburg.Georg Cantor's father was a Georg Cantor's father was a DanishDanish man of Lutheran religion. man of Lutheran religion.His mother, Maria Anna Böhm, was born in St. Petersburg and came from an His mother, Maria Anna Böhm, was born in St. Petersburg and came from an AustrianAustrian Roman Catholic family. Roman Catholic family.In 1860, Cantor graduated with distinction from the Realschule in Darmstadt; his In 1860, Cantor graduated with distinction from the Realschule in Darmstadt; his exceptional skills in mathematics, exceptional skills in mathematics, trigonometrytrigonometry in particular, were noted. In 1862, in particular, were noted. In 1862, following his father's wishes, Cantor entered the Federal Polytechnic Institute in following his father's wishes, Cantor entered the Federal Polytechnic Institute in Zurich, today the ETH Zurich and began studying mathematics.Zurich, today the ETH Zurich and began studying mathematics.In 1867, Berlin granted him the Ph.D. for a thesis on number theory, De In 1867, Berlin granted him the Ph.D. for a thesis on number theory, De aequationibus secundi gradus indeterminatis. After teaching one year in a Berlin aequationibus secundi gradus indeterminatis. After teaching one year in a Berlin girls' school, Cantor took up a position at the University of Halle, where he spent girls' school, Cantor took up a position at the University of Halle, where he spent his entire career. He was awarded the requisite habilitation for his thesis on number his entire career. He was awarded the requisite habilitation for his thesis on number theory.theory.

Cantor was promoted to Extraordinary Professor in 1872, and made full Professor Cantor was promoted to Extraordinary Professor in 1872, and made full Professor in 1879. To attain the latter rank at the age of 34 was a notable accomplishment, in 1879. To attain the latter rank at the age of 34 was a notable accomplishment, but Cantor very much desired a chair at a more prestigious university, in particular but Cantor very much desired a chair at a more prestigious university, in particular at Berlin, then the leading German university.at Berlin, then the leading German university.

. However, Kronecker, who headed mathematics at Berlin until his death in 1891, . However, Kronecker, who headed mathematics at Berlin until his death in 1891, and his colleague Hermann Schwarz were not agreeable to having Cantor as a and his colleague Hermann Schwarz were not agreeable to having Cantor as a colleague.colleague.

In 1890, Cantor was instrumental in founding the In 1890, Cantor was instrumental in founding the Deutsche Mathematiker-Deutsche Mathematiker-VereinigungVereinigung, chaired its first meeting in Halle in 1891, and was elected its first , chaired its first meeting in Halle in 1891, and was elected its first president. president.

In 1911, Cantor was one of the distinguished foreign scholars invited to attend the In 1911, Cantor was one of the distinguished foreign scholars invited to attend the 500th anniversary of the founding of the University of St. Andrews in 500th anniversary of the founding of the University of St. Andrews in ScotlandScotland. .

Cantor retired in 1913, and suffered from poverty, even hunger, during Cantor retired in 1913, and suffered from poverty, even hunger, during World War IWorld War I. The public celebration of his 70th birthday was cancelled because of . The public celebration of his 70th birthday was cancelled because of the war. He died in the sanatorium where he had spent the final year of his life. the war. He died in the sanatorium where he had spent the final year of his life.

He was the first to see that infinite sets come in different sizes, as follows. He first showed that He was the first to see that infinite sets come in different sizes, as follows. He first showed that given any set given any set AA, the set of all possible subsets of , the set of all possible subsets of AA, called the power set of , called the power set of AA, exists. He then proved , exists. He then proved that the power set of an infinite set that the power set of an infinite set AA has a size greater than the size of has a size greater than the size of AA (this fact is now known as (this fact is now known as Cantor's theorem). Thus there is an infinite hierarchy of sizes of infinite sets, from which springs Cantor's theorem). Thus there is an infinite hierarchy of sizes of infinite sets, from which springs the transfinite cardinal and the transfinite cardinal and ordinal numbersordinal numbers, and their peculiar arithmetic. His notation for the , and their peculiar arithmetic. His notation for the cardinal numbers was the Hebrew letter aleph with a natural number subscript; for the ordinals he cardinal numbers was the Hebrew letter aleph with a natural number subscript; for the ordinals he employed the Greek letter omega. employed the Greek letter omega. Cantor was the first to appreciate the value of one-to-one correspondences (hereinafter denoted "1-Cantor was the first to appreciate the value of one-to-one correspondences (hereinafter denoted "1-to-1") for set theory. He defined finite and infinite sets, breaking down the latter into denumerable to-1") for set theory. He defined finite and infinite sets, breaking down the latter into denumerable and nondenumerable setsThere exists a 1-to-1 correspondence between any denumerable set and the and nondenumerable setsThere exists a 1-to-1 correspondence between any denumerable set and the set of all natural numbers; all other infinite sets are nondenumerable. He proved that the set of all set of all natural numbers; all other infinite sets are nondenumerable. He proved that the set of all rational numbers is denumerable, but that the set of all real numbers is not and hence is strictly rational numbers is denumerable, but that the set of all real numbers is not and hence is strictly bigger. The cardinality of the natural numbers is aleph-null; that of the reals is larger, and is at least bigger. The cardinality of the natural numbers is aleph-null; that of the reals is larger, and is at least aleph-one (the latter being the next smallest cardinal after aleph-null). aleph-one (the latter being the next smallest cardinal after aleph-null). Cantor was the first to show that there was more than one kind of infinity. In doing so, he became Cantor was the first to show that there was more than one kind of infinity. In doing so, he became the first to invoke the notion of a 1-to-1 correspondence, albeit not calling it such. the first to invoke the notion of a 1-to-1 correspondence, albeit not calling it such. Between 1879 and 1884, Cantor published a series of six articles in Between 1879 and 1884, Cantor published a series of six articles in Mathematische AnnalenMathematische Annalen that that together formed an introduction to his set theory. together formed an introduction to his set theory. Cantor was the first to show that there was more than one kind of infinity. In doing so, he became Cantor was the first to show that there was more than one kind of infinity. In doing so, he became the first to invoke the notion of a 1-to-1 correspondence.the first to invoke the notion of a 1-to-1 correspondence.Cantor's work did attract favorable notice beyond Hilbert's celebrated encomium. In public lectures Cantor's work did attract favorable notice beyond Hilbert's celebrated encomium. In public lectures delivered at the first International Congress of Mathematicians, held in Zurich in 1897, Hurwitz and delivered at the first International Congress of Mathematicians, held in Zurich in 1897, Hurwitz and Hadamard both expressed their admiration for Cantor's set theory. Hadamard both expressed their admiration for Cantor's set theory.

Georg Ferdinand Ludwig Philipp CantorGeorg Ferdinand Ludwig Philipp Cantor

Born: 3 March 1845 in St Petersburg, RussiaDied: 6 Jan 1918 in Halle, Germany

Richard DedekindRichard Dedekind 's father was a professor at the Collegium Carolinum in 's father was a professor at the Collegium Carolinum in Brunswick. His mother was the daughter of a professor who also worked at Brunswick. His mother was the daughter of a professor who also worked at the Collegium Carolinum .the Collegium Carolinum .

He attended school in Brunswick from the age of seven .He attended school in Brunswick from the age of seven .

The Collegium Carolinum was an educational institution between a high The Collegium Carolinum was an educational institution between a high school and a university and he entered it in 1848 at the age of 16. school and a university and he entered it in 1848 at the age of 16.

In the autumn term of 1850, Dedekind attended his first course given by In the autumn term of 1850, Dedekind attended his first course given by GaussGauss. It was a course on least squares and [1]:-. It was a course on least squares and [1]:-

In 1854 both In 1854 both RiemannRiemann and Dedekind were awarded their habilitation and Dedekind were awarded their habilitation degrees within a few weeks of each other. Dedekind was then qualified as degrees within a few weeks of each other. Dedekind was then qualified as a university teacher and he began teaching at Göttingen giving courses on a university teacher and he began teaching at Göttingen giving courses on probability and geometry. probability and geometry.

The idea that came to him on 24 November 1858 was that every real number r divides the rational The idea that came to him on 24 November 1858 was that every real number r divides the rational numbers into two subsets, namely those greater than r and those less than r. Dedekind's brilliant idea numbers into two subsets, namely those greater than r and those less than r. Dedekind's brilliant idea was to represent the real numbers by such divisions of the rationals. was to represent the real numbers by such divisions of the rationals. The Collegium Carolinum in Brunswick had been upgraded to the Brunswick Polytechnikum by the The Collegium Carolinum in Brunswick had been upgraded to the Brunswick Polytechnikum by the 1860s, and Dedekind was appointed to the Polytechnikum in 1862 1860s, and Dedekind was appointed to the Polytechnikum in 1862 As well as his analysis of the nature of number, his work on mathematical induction, including the As well as his analysis of the nature of number, his work on mathematical induction, including the definition of finite and infinite sets, and his work in number theory, particularly in algebraic number definition of finite and infinite sets, and his work in number theory, particularly in algebraic number fields, is of major importance. fields, is of major importance. It was in the third and fourth editions of It was in the third and fourth editions of Vorlesungen über ZahlentheorieVorlesungen über Zahlentheorie, published in 1879 and , published in 1879 and 1894, that Dedekind wrote supplements in which he introduced the notion of an ideal which is 1894, that Dedekind wrote supplements in which he introduced the notion of an ideal which is fundamental to ring theory. Dedekind formulated his theory in the ring of integers of an algebraic fundamental to ring theory. Dedekind formulated his theory in the ring of integers of an algebraic number field.number field.He presented a logical theory of number and of complete induction, presented his principal He presented a logical theory of number and of complete induction, presented his principal conception of the essence of arithmetic, and dealt with the role of the complete system of real conception of the essence of arithmetic, and dealt with the role of the complete system of real numbers in geometry in the problem of the continuity of space. Among other things, he provides a numbers in geometry in the problem of the continuity of space. Among other things, he provides a definition independent of the concept of number for the infiniteness or finiteness of a set by using definition independent of the concept of number for the infiniteness or finiteness of a set by using the concept of mappingthe concept of mapping and treating the recursive definition, which is so important to the theory of and treating the recursive definition, which is so important to the theory of ordinal numbers. ordinal numbers. Dedekind's brilliance consisted not only of the theorems and concepts that he studied but, because of Dedekind's brilliance consisted not only of the theorems and concepts that he studied but, because of his ability to formulate and express his ideas so clearly, he introduced a new style of mathematics his ability to formulate and express his ideas so clearly, he introduced a new style of mathematics that been a major influence on mathematicians ever since. that been a major influence on mathematicians ever since.

Julius Wilhelm Richard DedekindJulius Wilhelm Richard Dedekind

Born: 6 Oct 1831 in Braunschweig,Germany Died: 12 Feb 1916 in Braunschweig,Germany

Archimedes of SyracuseArchimedes of Syracuse ( (GreekGreek: : ἈρχιμήδηςἈρχιμήδης) () (cc. 287 BC – . 287 BC – c.c. 212 BC) was a 212 BC) was a Greek Greek mathematicianmathematician, , physicistphysicist, , engineerengineer, , inventorinventor, and , and astronomerastronomer. . Archimedes is considered to be one of the greatest Archimedes is considered to be one of the greatest mathematiciansmathematicians of all time. of all time.[2][2] He used the He used the method of exhaustionmethod of exhaustion to calculate the to calculate the areaarea under the arc of a under the arc of a parabolaparabola with the with the summation of an infinite seriessummation of an infinite series, and gave a remarkably accurate approximation of , and gave a remarkably accurate approximation of PiPi..[3][3] He He also defined the also defined the spiralspiral bearing his name, formulas for the bearing his name, formulas for the volumesvolumes of of surfaces of revolutionsurfaces of revolution and an ingenious system for expressing very large numbersand an ingenious system for expressing very large numbers Archimedes was born Archimedes was born cc. 287 BC in the seaport city of . 287 BC in the seaport city of Syracuse, SicilySyracuse, Sicily, at that time a colony of , at that time a colony of Magna GraeciaMagna Graecia A biography of Archimedes was written by his friend Heracleides but this work has been lost, A biography of Archimedes was written by his friend Heracleides but this work has been lost, leaving the details of his life obscure.leaving the details of his life obscure.[8][8] It is unknown, for instance, whether he ever married It is unknown, for instance, whether he ever married or had childrenor had children Archimedes died Archimedes died cc. 212 BC during the . 212 BC during the Second Punic WarSecond Punic War, when Roman forces under General , when Roman forces under General Marcus Claudius MarcellusMarcus Claudius Marcellus captured the city of Syracuse after a two-year-long captured the city of Syracuse after a two-year-long siegesiege. . According to the popular account given by Plutarch, Archimedes was contemplating a According to the popular account given by Plutarch, Archimedes was contemplating a mathematical diagram when the city was captured. A Roman soldier commanded him to come mathematical diagram when the city was captured. A Roman soldier commanded him to come and meet General Marcellus but he declined, saying that he had to finish working on the and meet General Marcellus but he declined, saying that he had to finish working on the problem. The soldier was enraged by this, and killed Archimedes with his sword. problem. The soldier was enraged by this, and killed Archimedes with his sword.

While he is often regarded as a designer of mechanical devices, Archimedes also made While he is often regarded as a designer of mechanical devices, Archimedes also made contributions to the field of mathematics. contributions to the field of mathematics. Archimedes was able to use Archimedes was able to use infinitesimalsinfinitesimals in a way that is similar to modern in a way that is similar to modern integral calculusintegral calculus. . By assuming a proposition to be true and showing that this would lead to a By assuming a proposition to be true and showing that this would lead to a contradictioncontradiction, he , he could give answers to problems to an arbitrary degree of accuracy, while specifying the limits could give answers to problems to an arbitrary degree of accuracy, while specifying the limits within which the answer lay. This technique is known as the within which the answer lay. This technique is known as the method of exhaustionmethod of exhaustion, and he , and he employed it to approximate the value of employed it to approximate the value of ππ (Pi (Pi ) )In In The Measurement of a CircleThe Measurement of a Circle, Archimedes gives the value of the , Archimedes gives the value of the square rootsquare root of 3 as being of 3 as being more than 265/153 (approximately 1.7320261) and less than 1351/780 (approximately more than 265/153 (approximately 1.7320261) and less than 1351/780 (approximately 1.7320512). The actual value is approximately 1.7320508, making this a very accurate 1.7320512). The actual value is approximately 1.7320508, making this a very accurate estimate. estimate. In In The Quadrature of the ParabolaThe Quadrature of the Parabola, Archimedes proved that the area enclosed by a , Archimedes proved that the area enclosed by a parabolaparabola and a straight line is 4/3 multiplied by the area of a and a straight line is 4/3 multiplied by the area of a triangletriangle with equal base and height. He with equal base and height. He expressed the solution to the problem as a expressed the solution to the problem as a geometric seriesgeometric series that that summed to infinitysummed to infinity with the with the ratioratio 1/4: 1/4:If the first term in this series is the area of the triangle, then the second is the sum of the areas If the first term in this series is the area of the triangle, then the second is the sum of the areas of two triangles whose bases are the two smaller of two triangles whose bases are the two smaller secant linessecant lines, and so on. This proof is a , and so on. This proof is a variation of the variation of the infinite seriesinfinite series 1/4 + 1/16 + 1/64 + 1/256 + · · ·1/4 + 1/16 + 1/64 + 1/256 + · · · which sums to 1/3. which sums to 1/3.Archimedes devised a system of counting based on the Archimedes devised a system of counting based on the myriadmyriad. The word is from the Greek . The word is from the Greek μυριάς μυριάς muriasmurias, for the number 10,000. He proposed a number system using powers of a , for the number 10,000. He proposed a number system using powers of a myriad of myriads (100 million) and concluded that the number of grains of sand required to myriad of myriads (100 million) and concluded that the number of grains of sand required to fill the universe would be 8×1063, which can also be expressed as eight fill the universe would be 8×1063, which can also be expressed as eight vigintillionvigintillion. . [35][35]

Surviving worksSurviving works

On plane equilibriums, Quadrature of the parabola, On the On plane equilibriums, Quadrature of the parabola, On the sphere and cylinder, On spirals, On conoids and spheroids, On sphere and cylinder, On spirals, On conoids and spheroids, On floating bodies, Measurement of a circle, The Sandreckoner, floating bodies, Measurement of a circle, The Sandreckoner, On the method of mechanical problems.On the method of mechanical problems.

Place inPlace inHistoryHistory

Generally regarded as the greatest mathematician and scientist Generally regarded as the greatest mathematician and scientist of antiquity and one of the three greatest mathematicians of all of antiquity and one of the three greatest mathematicians of all time (together with time (together with Isaac Newton Isaac Newton (English 1643-1727) and (English 1643-1727) and Carl Friedrich Gauss (German 1777-1855)). Carl Friedrich Gauss (German 1777-1855)).

Archimedes of Syracuse

Born: 287 BC in Syracuse, SicilyDied: 212 BC in Syracuse, Sicily